Chapter ! Bell Shaped

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1 Chapter Business Statistics: A First Course 5 th Edition Chapter 7 Continuous Probability Distributions Learning Objectives In this chapter, you learn:! To compute probabilities from the normal distribution! To use the normal probability plot to determine whether a set of data is approximately normally distributed Business Statistics: A First Course, 5e 29 Prentice-Hall, Inc. Chap 6-1 BUS21: Business Statistics Continuous Probability Distributions - 2! Continuous random variable: How much a variable that can assume any value on a continuum (a line on a graph)! Note: Continuous Random Variable! These can potentially take on any value depending only on our ability to precisely and accurately measure (versus counting as we do in discrete random variables)!! thickness of an item! time required to complete a task! temperature of a solution! height, in inches!in continuous variables there is always another point in between any two!points on the line (in other words, any interval is infinitely divisible) BUS21: Business Statistics Continuous Probability Distributions - 3! Bell Shaped! Symmetrical! Mean Median are Equal Mode f() " Location is determined by the mean, µ " Spread is determined by the standard deviation,! " The random variable ranges to infinity in both directions BUS21: Business Statistics Continuous Probability Distributions - 4!" #"! The formula for the normal probability density function is Same mean, Different sigma f() = 1 2"# e$ 1 % ' 2 & ( $µ ) # 2 ( * ) Different mean, Same sigma Where e = the mathematical constant approximated by ! = the mathematical constant approximated by µ = the population mean " = the population standard deviation = any value of the continuous variable By varying the parameters µ and!, we obtain different normal distributions BUS21: Business Statistics Continuous Probability Distributions - 5 BUS21: Business Statistics Continuous Probability Distributions - 6

2 Chapter f() Changing µ shifts the distribution left or right. µ! Changing! varies the spread. As! gets larger, the spread gets wider.! Any normal distribution can be transformed into the standardized normal distribution ()! Need to transform units into units using the formula: = " µ # BUS21: Business Statistics Continuous Probability Distributions - 7 BUS21: Business Statistics Continuous Probability Distributions - 8! Units are in standard deviations! Mean =! Standard Deviation = 1 f() Standardized Normal Distribution Also known as the distribution Standardized Normal Distribution! Standardized normal probability density function f() = 1 2 2) e#(1/ 2" Values below the mean have negative -values 1 Values above the mean have positive -values Where e = the mathematical constant approximated by " = the mathematical constant approximated by = any value of the standardized normal distribution BUS21: Business Statistics Continuous Probability Distributions - 9 BUS21: Business Statistics Continuous Probability Distributions - 1 Distribution Example! is distributed normally with mean = 1 and standard deviation = 5,! What is the -value for = 2? = " µ 2 " 1 = = 2. # 5 This means that, for this distribution# is two standard deviations above the mean. Distribution Example Comparing and units: f() Original Units: Standardized Units: Notice that the shape of the distribution is the same, only the scale has changed. (! = 5) (! = 1) BUS21: Business Statistics Continuous Probability Distributions - 11 BUS21: Business Statistics Continuous Probability Distributions - 12

3 Chapter f()! Probabilities of the Normal Curve Probability is measured by the area under the curve P(a < < b) f() Probabilities of the Normal Curve Total area is 1., thus total probability = 1. Since the curve is symmetric, half is below the mean, half is above the mean Note:! the probability of any! individual value is zero!.5.5 a b! goes to negative infinity µ goes to positive infinity BUS21: Business Statistics Continuous Probability Distributions - 13 BUS21: Business Statistics Continuous Probability Distributions - 14 The Table! Cumulative table! gives the probability less than a desired value of! In other words, from negative infinity to The Table The column gives the value of to the second decimal point..1.2 # (cont.) Table E.2 is on" p. 546 of the book " Example: P( < 2.) =.9772 Note:! Before the Standard Table can be used, the value of must be converted to a value!! The row shows the value of to the first decimal point P( < 2.) =.9772 BUS21: Business Statistics Continuous Probability Distributions - 15 BUS21: Business Statistics Continuous Probability Distributions - 16 Example 1 Example 1! Let = the time it takes to download an image file from the internet.! Suppose has a normal distribution, with#! mean = 8., and! standard deviation = 5..! First, convert to : = " µ " 8. = = # 5. P( < ) P( < ) " What is the probability that is less than? µ = 8! = 5 µ =! = 1 Find P( < ) 8. BUS21: Business Statistics Continuous Probability Distributions - 17 BUS21: Business Statistics Continuous Probability Distributions - 18

4 Chapter Example 1 Example 2 From Probability Table P( < ) = P(=) =.5478! Let = the time it takes to download an image file from the internet.! has a normal distribution, with#! mean = 8., and! standard deviation = " What is the probability that is greater than? Find P( > ) 8. BUS21: Business Statistics Continuous Probability Distributions - 19 BUS21: Business Statistics Continuous Probability Distributions - 2 Example 2 P( > ) = P( > ) = 1 - P( $ ) thus = = BUS21: Business Statistics Continuous Probability Distributions - 21 Example 3! Let = the time it takes to download an image file from the internet.! has a normal distribution, with#! mean = 8., and! standard deviation = 5.. " What is the probability that is between 8. and? Find P(8. < < ) 8. BUS21: Business Statistics Continuous Probability Distributions - 22 Example 3 Example 3! First, convert both values to values: L = " µ # = 8 " 8 = U = " µ 5 # = " 8 = 5 P(8. < < ) P(. < < ) From Probability Table = P(. < < ) = P( < ) P( $.) = =.478 µ = 8! = 5 µ =! = µ =! = BUS21: Business Statistics Continuous Probability Distributions - 23 BUS21: Business Statistics Continuous Probability Distributions - 24

5 Chapter The Empirical Rule Revisited! For any Normal Distribution! Approximately 68% of the data is within 1 standard deviation of the mean The Empirical Rule Revisited! Approximately 95% of the data lies within two standard deviations of the mean, or % ± 2!! Approximately 99.7% of the data lies within three standard deviations of the mean, or % ± 3! 68.27% " " µ µ ±1" 95.45% µ ± 2" Remember:!!this is only true for a! bell-shaped distribution! 99.73% µ ± 3" BUS21: Business Statistics Continuous Probability Distributions - 25 BUS21: Business Statistics Continuous Probability Distributions - 26 Example Problem 4! Let = the time in seconds it takes to download an image file from the internet.! has a normal distribution, with#! mean = 8., and! standard deviation = 5.. " What value of gives 2% of download times less than? Find P() $.2 =? 8. This area =.2 BUS21: Business Statistics Continuous Probability Distributions From Probability Table Example Problem 4.5! 2% area in the lower tail is consistent with a value of ? BUS21: Business Statistics Continuous Probability Distributions - 28! Convert units back to units:! Therefore, Example Problem 4 since = - µ then = µ + " " so = µ + " = 8. + (#.84)5. = 3.8 2% of the time, it takes less than 3.8 seconds to download an image (assuming a distribution with mean of 8 and standard deviation of 5) Example: Budget Auto Budget Auto sells a popular antifreeze. When the stock of antifreeze drops to 2 gallons, a replenishment order is placed. The store manager is concerned that sales are being lost due to running out of product while waiting for an order. Demand during replenishment lead-time is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons. The manager would like to know the probability of a stockout, P(x > 2). BUS21: Business Statistics Continuous Probability Distributions - 29 BUS21: Business Statistics Continuous Probability Distributions - 3

6 Chapter Example: Budget Auto Solving for the Stockout Probability Step 1: Convert x to the standard normal distribution. = - µ " = Step 2: Find the area under the standard normal curve to the left of z =.83. see next slide =.83 Cumulative Probability Table for the Standard Normal Distribution z P(z <.83) BUS21: Business Statistics Continuous Probability Distributions - 31 BUS21: Business Statistics Continuous Probability Distributions - 32 Solving for the Stockout Probability Step 3: Compute the area under the standard normal curve to the right of z =.83. P(z >.83) =1" P(z #.83) =1".7967 =.233 Solving for the Stockout Probability Area =.7967 Area = =.233 Probability of a stockout.83 z BUS21: Business Statistics Continuous Probability Distributions - 33 BUS21: Business Statistics Continuous Probability Distributions - 34 Standard Solving for the Reorder Point If the manager of Pep one wants the probability of a stockout to be no more than.5, what should the reorder point be? Area =.95 Area =.5 z.5 z BUS21: Business Statistics Continuous Probability Distributions - 35 BUS21: Business Statistics Continuous Probability Distributions - 36

7 Chapter Solving for the Reorder Point Step 1: Find the z-value that cuts off an area of.5 in the right tail of the std normal distribution.. z This.9418 gives.9429 us a value of We look up the complement of the.9744 tail area.975 ( =.95) Solving for the Reorder Point Step 2: Convert z.5 to the corresponding value of x. = µ +.5 " = 15 + (1.645)6 = # 25 By increasing the reorder point from 2 gallons to 25 gallons on hand, the probability of a stockout decreases from about.2 to.5. BUS21: Business Statistics Continuous Probability Distributions - 37 BUS21: Business Statistics Continuous Probability Distributions - 38 Chapter Summary! Presented normal distribution! Found probabilities for the normal distribution! Applied normal distribution to problems BUS21: Business Statistics Continuous Probability Distributions - 39